Inhomogeneous Poisson process

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In probability theory, an inhomogeneous Poisson process (or non-homogeneous Poisson process) is a Poisson process with rate parameter $\lambda (t)$ such that the rate parameter of the process is a function of time.[1] Inhomogeneous Poisson processes have been used to describe numerous random phenomena[2] including cyclone prediction,[3] arrival times of calls to a call centre in a hospital laboratory[4] and call centre,[5] arrival times of aircraft to airspace around an airport[6] and database transaction times.[7]

The Cox process is an extension of this model where λ(t) itself can be a stochastic or random process.

Definition

Write $N(t)$ for the number of events by time $t$. A stochastic process is an inhomogeneous Poisson process for some small value $h$ if:[1][8]

1. $N(0)=0$
2. Non-overlapping increments are independent
3. $P(N(t+h)-N(t)=1) = \lambda(t) h + o(h)$
4. $P(N(t+h)-N(t)>1) = o(h)$

for all t and where, in big o notation, $\scriptstyle \frac {o(h)}{h} \rightarrow 0\; \mathrm{as}\, h\, \rightarrow 0$. In the case of point processes with refractoriness (e.g., neural spike trains) a stronger version of property 4 holds:[9] $P(N(t+h)-N(t)>1) = o(h^2)$.

Properties

Write N(t) for the number of events by time t and $\scriptstyle m(t) = \int_0^{t} \lambda (u)\text{d}u$ for the mean. Then N(t) has a Poisson distribution with parameter m(t), that is for k = 0, 1, 2, 3….[10]

$\mathbb P(N(t)=k) = \frac{m(t)^k}{k!}e^{-m(t)}.$

Fitting

Traffic on the AT&T long distance network was modelled by a inhomogeneous Poisson process with piecewise linear rate function.[11] Ordinary least squares, iterative weighted least squares and maximum likelihood methods were evaluated and maximum likelihood shown to perform best overall for the data.

Simulation

An inhomogeneous Poisson process with intensity function λ(t) can be simulated by rejection sampling from a homogeneous Poisson process with fixed rate λ: choose a sufficiently large λ so that λ(t) = λ p(t) and simulate a Poisson process with rate parameter λ. Accept an event from the Poisson simulation at time t with probability p(t).[1][12] For a log-linear rate function a more efficient method was published by Lewis and Shedler in 1975.[13]

References

1. ^ a b c Ross, Sheldon M. (2006). Simulation. Academic Press. p. 32. ISBN 0-12-598063-9.
2. ^ Leemis, Larry (May 2003). "Estimating and Simulating Nonhomogeneous Poisson Processes" (PDF). William and Mary Mathematics Department. Retrieved Sep 26, 2011.
3. ^ Lee, Sanghoon; Wilson, James R.; Crawford, Melba M. (1991). "Modeling and simulation of a nonhomogeneous poisson process having cyclic behavior" (PDF). Communications in Statistics - Simulation and Computation 20 (2-3): 777–809. doi:10.1080/03610919108812984.
4. ^ Kao, Edward P. C.; Chang, Sheng-Lin (November 1988). "Modeling Time-Dependent Arrivals to Service Systems: A Case in Using a Piecewise-Polynomial Rate Function in a Nonhomogeneous Poisson Process". Management Science (INFORMS) 34 (11): 1367–1379. doi:10.1287/mnsc.34.11.1367. JSTOR 2631999.
5. ^ Weinberg, J.; Brown, L. D.; Stroud, J. R. (2007). "Bayesian Forecasting of an Inhomogeneous Poisson Process with Applications to Call Center Data". Journal of the American Statistical Association 102 (480): 1185. doi:10.1198/016214506000001455. edit
6. ^ Galliher, Herbert P.; Wheeler, R. Clyde (March–April 1958). "Nonstationary Queuing Probabilities for Landing Congestion of Aircraft". Operations Research 6 (2): 264–275. doi:10.1287/opre.6.2.264. JSTOR 167618.
7. ^ Lewis, P. A. W.; Shedler, G. S. (September 1976). "Statistical Analysis of Non-stationary Series of Events in a Data Base System". IBM Journal of Research and Development 20 (5). doi:10.1147/rd.205.0465. CiteSeerX: 10.1.1.84.9018.
8. ^ Srinivasan (1974). "Chapter 2". Stochastic point processes and their applications. ISBN 0-85264-223-7.
9. ^ L. Citi, D. Ba, E.N. Brown, and R. Barbieri (2014). "Likelihood methods for point processes with refractoriness". Neural Computation. doi:10.1162/NECO_a_00548.
10. ^ Pham, Hoang (2006). "Software Reliability Modeling". System Software Reliability. Springer Series in Reliability Engineering. pp. 153–177. doi:10.1007/1-84628-295-0_5. ISBN 978-1-85233-950-0. edit
11. ^ Massey, W. A.; Parker, G. A.; Whitt, W. (1996). "Estimating the parameters of a nonhomogeneous Poisson process with linear rate" (PDF). Telecommunication Systems 5 (2): 361. doi:10.1007/BF02112523. edit
12. ^ Lewis, P. A. W.; Shedler, G. S. (1979). "Simulation of nonhomogeneous poisson processes by thinning" (PDF). Naval Research Logistics Quarterly 26 (3): 403. doi:10.1002/nav.3800260304. edit
13. ^ Lewis, P. A. W.; Shedler, G. S. (1976). "Simulation of nonhomogeneous Poisson processes with log linear rate function". Biometrika 63 (3): 501. doi:10.1093/biomet/63.3.501. JSTOR 2335727. edit